Quadratic-phase Laplace transform: definition and properties

Eigenfunctions of a non-self-adjoint second-order differential operator T with complex coefficients are typically endowed with an intrinsic quadratic phase structure, which prevents classical Laplace and Fourier transforms from effectively characterizing its spectral properties. In this paper, a class of quadratic-phase Laplace transforms (QPLT) is constructed from differential operators with quadratic phase structures, whose kernel is shown to be the unique solution of the spectral problem associated with the operator T. Subsequently, the inversion formula and the Parseval identity for the QPLT are established, and its operational properties, including shifting, scaling, differentiation, convolution, and correlation theorems, are derived. Finally, the Donoho–Stark uncertainty principle for the QPLT is proved. The QPLT is demonstrated to provide an effective analytical tool for the study of non-self-adjoint quadratic operators and the associated partial differential equations.